RMS-Speed Scaling with Temperature and Molar Mass
For an ideal gas, v_rms = sqrt(3RT/M); hence speed ratios depend on sqrt(T/M) and do not directly depend on gas pressure.
Why this shows up in the exam
Identifying a gas from rms speed · Comparing hydrogen, oxygen, and noble gases · Dissociation and temperature-change questions
Learn the idea
Molecular thermal speed grows as square root of absolute temperature and falls as square root of molar mass. Heating makes every species faster, while at the same temperature a lighter species must move faster to carry the same average translational energy.
🧠 Memory hook: Speed follows the square root of T over M.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_rms = sqrt(3RT/M) — rms speed using SI molar mass M for an ideal gas
- v2/v1 = sqrt((T2 M1)/(T1 M2)) — ratio rule for two gas states or species
How to approach it
- 1Write the ratio before inserting values
- 2Convert every temperature to kelvin
- 3Square only after isolating the desired ratio
Common slip-ups that cost marks
- •Using Celsius temperature
- •Making rms speed proportional to pressure
- •Forgetting molecular mass changes after dissociation
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas has rms molecular speed 300 m/s at 300 K. What is its rms speed at 1200 K, assuming ideal behavior?
More from Kinetic Theory of Gases
Ideal gas law and gas laws
The ideal gas law and related gas laws describe the relationships between pressure, volume, temperature, and number of moles for ideal gases.
Degrees of freedom and thermal properties
Degrees of freedom determine the distribution of energy among molecules, affecting internal energy, specific heats, and the ratio of specific heats (γ).
Kinetic theory and molecular motion
The kinetic theory explains the behavior of gases in terms of the motion and collisions of their molecules, relating properties like pressure, temperature, and kinetic energy.
RMS speed and temperature dependence
The root mean square (rms) speed of gas molecules depends on temperature and molar mass, and is a key measure of molecular motion in gases.
Mean free path and collisions
Mean free path is the average distance a molecule travels between collisions, and depends on molecular size and number density.
Ideal-Gas Equation and Molecular Form
For a dilute ideal gas in thermal equilibrium, the state variables satisfy PV = nRT = Nk_B T, where intermolecular potential energy and molecular volume are neglected.